The Comparison Of Numerical Methods For Slab Optical Waveguide Discontinuities

Discontinuity problems are frequently encountered in practical applications of optical waveguides. In this thesis, based on scattering operators, we mainly deal with single and multiple discontinuities in planar optical waveguides.Firstly, in order to accurately model the reflections at the longitudinal discontinuous interfaces, we decompose the wave field u as u = u⁺ + u^- and get two one-way Helmholtz equations: ?zu⁺ = iLu⁺, ?_zu^- = —iLu^-, where L is the square root operator. Then, we introduce scattering operators R and T and develop a new non-iterative bidirectional beam propagation methods (BiBMPs) based on the two operators. As a consequence, we can solve the model by matching the two operators plus some boundary conditions.Secondly, as we all know, most wave propagation problems are posed in an infinite domain. In this case, we need to truncate the domain to obtain a finite domain for performing numerical calculations. For this purpose, we introduce the so-called Perfectly Matched Layers (PMLs) to truncate unbound domains. Besides, PMLs can theoretically absorb without reflection any wave travelling towards boundaries. Therefore, it is possible to have very accurate wave field distribution computed in the domain concerned.Thirdly, when the transverse operator ?_x²+ k₀²n²(x) is approximated by a tridi-agonal matrix A, we first use eigenvalue decomposition method (EDM) to derive an comparatively accurate evaluation of the square root operator L and the propagaor P. As a result, we then can march the reflection operator R and transmission operator T to accurately compute the reflected and transmitted wave field.However, when the order of A is large, EDM is time consuming and memory intensive. In order to reduce computational cost, we introduce modified rational [p/p] Pade approximant to approximate the square root operator L and [(q — 1)/q] Pade ap-proximant to approximate the propagator P in chapter 5, and consequently achieve another numerical method for BiBPM model. At the same time, the accuracy and computational cost of numerical results derived from the EDM method and the improved rational Pade approximation method are compared, which shows that the latter is more efficient for solving BiBPM model.Finally, for periodic waveguides, we develop a new method called truncated eigen-function expansion approach. This method can reduce the computational cost of every marching step in EDM while keeping high accuracy of the computed wave fields. So, it is a very efficient method for periodic waveguide structures.